Construct a field of 9 elements – construct the addition and multiplication tables. Begin with polynomials having coefficients $0$, $1$, and $2$ (integers modulo $3$) and use the modulus $X^2+X+2$
I really am not sure what its asking. I know how to do the multiple and addition of a field of $9$ but not sure about the rest at all.
Any help? Or even a similar example? Help me understand. Thanks!
Am I being asked to mod the addition and multiplication tables by $\ X^2+X+2 \quad (8) \ $? Am I supposed to write out all possible polynomials with modulo $3$ coefficients and $\mathrm{mod} \ X^2+X+1$?
Make no sense to me.
Since $x^2+x+2=0$ it follows that also $x^2=-x-2\equiv 2x+1$ since working mod 3. The elements of the field are only the nine linear terms $ax+b$ where each of $a,b$ can be independently any of $0,1,2$ giving the desired total of $9$ elements. Addition is no problem since the overflow can be done componentwise, e.g. $2x+2+x+0=3x+2=0x+2.$
For multiplication you use the above $x^2=2x+1$ when needed, for example $$(2x+1)\cdot (x+2)=2x^2+5x+2=2x^2+2x+2,\tag{1}$$ since $5=2$ mod 3, and now put in $2x+1$ for the $x^2$ so that $2x^2=4x+2=x+2$ replaces the $2x^2$ term in $(1)$, so it comes out $(x+2)+2x+2=3x+4=3x+1=0x+1.$
Other multiplications are similar, using the $x^2=2x+1$ as needed.